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Bottom lane matchups are the single most complex thing in League of Legends earlygame period. A 2v2 lane allows for a lot more different setups than a 1v1, and thus requires another level of thinking and planning pre-game to execute perfectly.
In this article I will not cover every match up extensively, but rather provide you with some information and thoughts that will help you plan out your bot lane strategy the next time you play a ranked match.Power spikes
The first concept we will cover is something you are probably familiar with if you enjoy reading about the game: powerspikes. To understand powerspikes, first you need to think about how every champion gains strength in League of Legends: by levelling up. However, not every champion gains the same power from every level up. A good first example are champions with two CC abilities, or abilities that usually work in a combo. Talking about supports, this would be Alistar, Blitzcrank, Leona and Thresh. These champions get a very powerful CC at level 1, and they get another very powerful CC that compliments the first at level 2. However, at levels 3, 4 and 5 they do not gain as much power as they do at 1 and 2, as they only get extra damage.
This means you have to be wary of levels 1 and 2 when playing as and against these champions, as this is where they are at the strongest point they will be for a while. If you are playing as them, this is the time to go for a play, and if you are facing them you should be wary of their engage unless you think you can win the fight.
Now obviously, gaining a second ability increases a lot of champions power by a big amount. Some champions, however, gain more relative strength from a different powerspike in lane. I'm referring to the champions that rely on one strong skill, like Graves (Q), Janna (E), Corki (Q), Morgana (Q), and the like. While these champions benefit a lot from their second ability at 2, especially in cases like Graves who gains mobility, they are at their strongest at every odd level until 9, that is, every level in which they get to put another point in their bread-and-butter strongest skill. You should look for fights when playing these champions as soon as you get levels 3, 5, 7 and 9, especially if you get it before the enemies do. The opposite applies once again: play carefully around the level 5 Graves if you are only level 4 as you will likely lose.
The last point of relevance to this subject is, as you might have guessed, level 6. Again, every champion is obviously very happy to get their ultimates. However some ultimates are enormously more impactful in lane than others. Graves+Annie is going to DEFINITELY want to fight Lucian+Nami at level 6. Generally, champions with strong level 6 powerspikes get strong burst damage from their ults (Graves, Jinx, Ezreal), hard CC (Leona, Braum, Sona) or both (Zyra, Annie, Morgana). Mage-type supports become VERY strong at level 6 due to their bursty nature, be especially careful around level 6 Zyra, Annie, Morgana, Vel'koz and the like.
Trading and lane dominance

One of the most important decisions you have to take when playing bot lane is when to trade and when to try for the all-in. Some matchups are pretty simple, such as Graves Leona vs. Ezreal Nami. The Graves lane should try to force all ins and dodge Ezreal Q's, while trying not to do even trades as they will get outsustained. The Ezreal lane has to dodge Leona engages, especially on Nami who has no escapes, and they need to try for even health trades to attempt to outsustain and eventually push the Graves Leona out of lane.
Usually at any point in time during 2v2 bot lane, one team can win the all-in if it happens right now. If you are in that position, you have lane dominance, so you can play aggresive: punish your opponent when he goes to CS, try to take over their brush, and look for the all-in. Basically, if you are positive you can win the engage, you should go look for it, it's not like the opponent can take the fight. This is usually the position that Graves, Lucian or Draven lanes are at.
However, other ADCs might outrange you with their poke: Ezreal Q, Jinx rockets, Kog'maw W's, Caitlyn, Tristana after a few levels. The difference in range will let them, if you do not space carefully, get free hits on you and wither you down until you no longer have lane dominance. The way to avoid this is by trying to close the gap: try not to get hit from outside your range. Dash forward, wait until your support lands a CC, hide behind minions if applicable.
This will be all I will cover in this piece. Next time you play bot lane, think about your champions strengths and powerspikes relative to the enemies, and try to plan your engages and trades around that. You will notice the difference!